We consider the Kane-Mele model in a ribbon geometry of a two-dimensional honeycomb lattice with zigzag edges, which is shown in Fig.REF  (a). The Hamiltonian of the Kane-Mele model is [1]}, [2]}:
\(\mathcal {H}_{\text{KM}} &=&t\sum _{\langle i,j\rangle ,\alpha }c_{i\alpha }^{\dagger }c_{j\alpha }+i\lambda _{\text{SO}}\sum _{\langle \langle i,j\rangle \rangle ,\alpha \beta }v_{ij}c_{i\alpha }^{\dagger }s^z_{\alpha \beta }c_{j\beta } \\&+&i\lambda _{\text{R}}\sum _{\langle i,j\rangle ,\alpha \beta }c_{i\alpha }^{\dagger }(\mathbf {s}\times \mathbf {d}_{ij})^z_{\alpha \beta }c_{j\beta }+\lambda _\nu \sum _{i\alpha }\xi _{i}c_{i\alpha }^{\dagger }c_{i\alpha }. \) 
